I have seen contradicting sources on whether building a succah is part of the mitzva to dwell in it. I want to know if there is a machlokes on whether building a succah is preparation of a mitzva or a mitzva in and of itself.
The Yerushalmi (Sukka 1:2) and Tosefta (Brachos 6:9) write that one should make a bracha on 'the mitzvah of building a sukkah'. However, the Bavli (Sukka daf 46a and see Rosh there) disagrees.
The simplest way to explain the dispute would be to say that the Yerushalmi believes that there's a mitzvah to build the sukkah, and the Bavli disagrees. I believe that the Rambam understood the dispute in this way, as he writes in a Teshuva (142) המצוה היא ישיבת הסוכה לא עשייתה ואין אנו עושין אלא כדי לישב, ולכן נברך לישב ולא נברך לעשות - the mitzvah is only to dwell in the Sukkah, not to make it, which is why the bracha is phrased as "לישב" instead of "לעשות". However, R. Zzi Pesach Frank (Mikraei Kodesh Sukka I, pg 111) explains that the Yerushalmi merely believes that even a preparation for a mitzvah requires a blessing, and I believe that this is the opinion of the Or Zarua (Hil. Tefilin 583) as well.
However, there are several indications that even according to the Bavli or how we'd pasken, there is still a mitzvah of sorts to build the sukkah:
The Mishnah (Shevuos 3:8) gives examples of someone who swore not to do a mitzvah, and includes someone who swore that he won't make a sukkah
Rashi to Makkos (8a) writes that there is a mitzvah to build a sukkah, but that one would be exempt from doing so if he found one already made. (This is noted by the Aruch L'ner (there) but the Avnei Nezer O.C. 459 disagrees with this reading of the Gemara)
Rama, quoting Maharil, writes that one should build the sukka immediately after Yom Kippur, because מצוה הבא לידך אל תחמיצנה - one shouldn't delay a mitvah (referring to building the Sukkah as a mitzvah)
Thus, it appears that building a sukka is a quasi-mitzvah, in that some rules of 'mitzvos' apply but some do not. The Netziv clarifies this in his commentary to the She'iltos of Rav Achai Gaon (169:1), where the She'iltos writes מצוה למעבד מטללתא ולמיתב בה - there is a mitzvah to make a sukkah and live in it. The Netziv explains, that while the main part of the mitzvah is the dwelling in it, and building a sukkah is merely a הכשר מצוה - a preparation for the mitzvah, there is significance to the fact that this is 'a preparation that is written in the Torah'. (I assume that he means וחג הסוכות תעשה לך, which, while al pi pshat seems to be referring to the holiday itself, is taken by the Gemara daf 11a to refer to the sukkah structure as well).
Yes, as HaRav Eliyahu Mansour explains (citing specific poseqim), the poseqim (Halakhic authorities) are divided on whether building a sukkah is a mitzwah itself or simply a preparatory activity for a mitzwah. He summarizes that:
"[...] There is a Machloket if building the Succah is a Mitzvah itself. We hold that it is a preparatory Mitzvah. There is no Beracha in the building, but one should involve himself in the construction. Specifically, try to begin on Motzae Yom Kippur to build the Succah or start learning the Halachot in regards to the Succah."
The reason there is a minhag to build Sukkot the day after yom kippur is to start the year off immediately with mitzvoth. Due to this reasoning, it is quite clear that building a sukkah is a mitzvah, or this reasoning would not be applicable.